Creating a Physics Problem with Coulomb's Law and Centripetal Force

[Shmi]

The idea being that I have two particles of opposite charge (particles implying the mass is negligible relative to the electric forces at play). One is fixed at the center, the other orbits around at a fixed radius. How would I go about solving for the angular velocity required to keep this system in a circular orbit when centripetal force definitions are riddled with mass terms?

 

[da_steve]

just an idea but if you have two particles with negligible mass, in Newtonian physics if they experienced a force they would have infinite acceleration >> they have to be infinitely far away

if you add some more conditions like speed of light etc ... well mass will eventually become significant

if you make them wave particles and use de broglies momentum of a wave ... maybe, but i don't think u'll find an answer other than infinity in classical physics

 

[Redbelly98]

You do need to define a mass for the orbiting particle, in order to relate the force acting on it and its motion (more specifically, its acceleration).

Also, we would need to assume that the "stationary" particle is much more massive than the orbiting one. It might help to think in terms of a planet orbiting the sun, or of Bohr's hydrogen atom model.

EDIT added:

...the mass is negligible relative to the electric forces at play...

By the way, this statement makes little sense. A mass can be negligible only relative to another mass, not relative to a different physical concept like force, distance, time, etc.

 

[Shmi]

Apologies, I've worked with this problem some more and understand the units better.

To keep this system at a constant radius (which I give to be ~Bohr radius of hydrogen atom), I want to find out how fast a classical particle would have to move to maintain this radius.

Am I crazy to equate the centripetal force and the coulomb's force? Here the units clearly match.

$$\frac{m_e v^2}{r} = \frac{kq_e^2}{r^2}$$

Solving for v, I get:

$$v = \sqrt{\frac{k q_e^2}{m_e r}}$$

Which for given values produces a number around 2.1*10^6 m/s. Reasonable? More importantly, is this a reasonable problem to solve for other physics students fresh out of mechanics and just learning early electrostatics?

 

[Redbelly98]

Am I crazy to equate the centripetal force and the coulomb's force?

Not crazy at all. That is exactly how centripetal force works; you equate it with the net force acting perpendicular to the particle's velocity.

By the way, you are basically solving the Bohr model of the hydrogen atom, except for the part where only discrete (quantized) orbits are allowed. I remember that the velocity of the ground state electron in Bohr's model is about 2 orders of magnitude slower than c, so your result looks reasonable.

 

[BruceW]

As I'm sure everyone already knows, its not so simple in quantum mechanics. But the equation does look correct for a classical particle.

 

[jtbell]

Am I crazy to equate the centripetal force and the coulomb's force?

No, because the centripetal force is the Coulomb force in this situation. "Centripetal force" isn't some new kind of force, it's simply a descriptive term that can be applied to any kind of force under the right circumstances, namely if the force always acts towards a central point. It could be the electrostatic force as in this situation, it could be the gravitational force, it could be the tension force exerted by a string, etc.

 

[Shmi]

Redbelly98 and jtbell, thank you for the guidance! I submitted the problem, and my professor liked it. Thanks.